# Full text of "Analytical Mechanics"

## See other formats

IMPULSE AND MOMENTUM 251 11. Two spheres of masses m and 2 m moving with equal velocities along two lines at right angles to each other collide at the instant when their centers are on the line of motion of the smaller sphere. Show that if the contact is smooth and e = 0.5 the smaller sphere will come to rest, and find the direction and magnitude of the velocity of the larger sphere. 12. In the preceding problem let m = 500 gm, v = 40 cm. per second and find (a) the impulse, its magnitude and direction; (b) the loss of energy. 13. A metal patched bullet which weighs 1.5 ounces strikes a rock, normally, with a velocity of 1500 feet per second. Find the velocity with which it will rebound and the impulse given to the rock; e — 0.5. 14. A body impinges against another body which has a mass n times as large. Show that if the larger body is at rest and the contact inelastic the loss of energy is —— times its value before the collision. 71 + 1 15. A particle is projected up a smooth inclined plane with a velocity \/rgh') simultaneously a particle of equal mass is allowed to slide down the inclined plane. The two collide somewhere on the plane. Find the velocities with which the particles will arrive at the bottom of the plane* h = the height of the inclined plane. 16. Two small spheres of masses m and 2 m move in a smooth circular groove on a horizontal table with equal speeds in opposite directions. Find the position of the second collision relative to the first; e = 0.6. 17. In the preceding problem find the interval of time between the first and the eleventh collision, under the following assumptions — the radii of the particles are negligible compared with that of the circular groove, which equals 50 cm., the common speed of the spheres just before the first collision is 500 cm. per second, the time of collision is negligible. 202. Efficiency of a Blow. — A blow may be struck to produce one or the other of two distinct results. The object of a blow from a hammer in driving a nail is quite different from that of a blow in shaping a rivet. Efficiency in the first case means greatest amount of driving with the least amount of deformation, while in the second case it means greatest amount of smashing with the least amount of driving. Therefore the efficiency of a blow is different for these two cases. We may define